Nakamura's epsilon-isomorphism conjecture for relative Robba rings

Let $M$) be a $(\varphi,\Gamma)$-module over a relative Robba ring, and let $\mathbf{1}$ denote the unit object in the associated determinant category. The fundamental line attached to $M$ is the determinant-category object associated with $M$.

Nakamura's conjecture. For every such $M$, there exists an isomorphism

$$\mathbf{1}\simeq \text{the fundamental line attached to }M.$$

This isomorphism should be compatible with the algebraic functional equation, duality, and the de Rham isomorphism. It is a relative form of the epsilon-isomorphism conjecture, motivated by applications to $p$-adic local Langlands and the local Tamagawa number conjecture; the source does not state a resolution.